
where \(b\)  and \(c\)  are positive constants. The above result is for a Minkowski background (\(\Lambda _{eff}=0\) ) but where Einstein gravity and a non-minimal coupling term acts on the vortex. The exponential fall-off expressions (REF ) and () are similar to those found in fixed Minkowski spacetime [1]} and we recover them if we set \(\xi =0\)  and \(D=1\) .
